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On clogged and fast diffusions in porous media with fractional pressure

Antonin Chodron de Courcel

TL;DR

This work analyzes weak solutions to a nonlocal porous-medium equation with fractional pressure on the torus, focusing on degenerate mobility regimes: clogged diffusion with mobility $\mu^{-m}$ for $m>0$ and fast diffusion with mobility $\mu^{m}$ for $m\in(0,1)$. Employing maximum principles for the fractional Laplacian, the authors derive instantaneous $L^\infty$ regularization, explicit lower barriers governed by ODEs, and a robust energy/dissipation framework. Existence is obtained through a viscosity-regularized approximation, with uniform a priori estimates and compactness culminating in weak solutions that enjoy mass conservation, a weak maximum principle, and infinite-speed propagation. The results illuminate how nonlocal interactions and fractional pressure influence propagation and regularization, and they connect to self-similar behavior (type I/II) and the broader theory of density-dependent diffusion in porous media.

Abstract

We study the existence and infinite-speed propagation of solutions to models arising in porous media, when the mobility is highly degenerate (inverse power law). The approach is based on maximum principles for the fractional Laplacian, and allows to derive lower bounds on solutions in a straightforward manner. Finally, in the case of clogged porous media, where the mobility vanishes at points of unbounded density, solutions that become instantaneously bounded are constructed.

On clogged and fast diffusions in porous media with fractional pressure

TL;DR

This work analyzes weak solutions to a nonlocal porous-medium equation with fractional pressure on the torus, focusing on degenerate mobility regimes: clogged diffusion with mobility for and fast diffusion with mobility for . Employing maximum principles for the fractional Laplacian, the authors derive instantaneous regularization, explicit lower barriers governed by ODEs, and a robust energy/dissipation framework. Existence is obtained through a viscosity-regularized approximation, with uniform a priori estimates and compactness culminating in weak solutions that enjoy mass conservation, a weak maximum principle, and infinite-speed propagation. The results illuminate how nonlocal interactions and fractional pressure influence propagation and regularization, and they connect to self-similar behavior (type I/II) and the broader theory of density-dependent diffusion in porous media.

Abstract

We study the existence and infinite-speed propagation of solutions to models arising in porous media, when the mobility is highly degenerate (inverse power law). The approach is based on maximum principles for the fractional Laplacian, and allows to derive lower bounds on solutions in a straightforward manner. Finally, in the case of clogged porous media, where the mobility vanishes at points of unbounded density, solutions that become instantaneously bounded are constructed.
Paper Structure (10 sections, 13 theorems, 114 equations)

This paper contains 10 sections, 13 theorems, 114 equations.

Table of Contents

  1. Introduction.
  2. Related works.
  3. The constant mobility case.
  4. Density-dependent mobility.
  5. Main results.
  6. Acknowledgments.
  7. Key estimates.
  8. Clogged diffusions.
  9. Existence theory.
  10. Lower barrier for fast diffusions.

Key Result

Theorem 1.2

Let $d\ge 1$, $(d-2)_+ < s<d$, and $m>0$. Define If $\gamma \ge 1$, consider $\mu_0 \in L^{p}({\mathbb{T}}^d)\cap \dot H^\frac{s-d}{2}({\mathbb{T}}^d)$, for some $p>\gamma$. Otherwise, we can take $\mu_0\in \mathcal{M}_+({\mathbb{T}}^d)\cap \dot H^\frac{s-d}{2}({\mathbb{T}}^d)$. Assume finally $\mu_0\ge 0$. Then, there exists a weak solution $\mu$

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2: Clogged diffusion
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Fast diffusion
  • Remark 1.9
  • Remark 1.10
  • ...and 34 more